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# Connected Particles

In mechanics, particles connected together by a string are usually associated with pulleys. These types of problems in mathematics realistically address how cars can be towed or how pulleys work with regard to tension.

## Calculating tension in connected particles

We consider particles separately in a system that involves multiple particles in motion. We can treat the system as a whole if particles are moving in the same straight line. In problems involving pulleys where particles are moving in different directions, the system will not be treated as a whole and you will need to resolve the mass of the particles separately.

Two particles A and B of mass 5kg and 2kg are connected by a light inextensible string on a smooth surface. When a horizontal force of XN gets applied to A in the direction away from B, the particle moves with an acceleration of

1. Find the value of X.

2. Find the tension in the string.

Connected particles

We will consider the whole diagram as one and treat it as such dealing with the equations. When you carefully examine the next figure, you will realise that though both particles have tension associated with them, particle B does not have force XN to account for. This is why we will look at the system as a whole.

Draw an extension indicating all forces acting on the particles. Though gravity is not relevant to this question, we will still include it in our diagram.

Forces acting on the particles

1.

2. This time we can just look at particle B.

## Connected particles and pulleys

Let's look at problems involving two particles connected by a light inextensible string passing over a fixed pulley. Pulleys are simple machines that are designed to help lift loads. The design involves a rope looped over a wheel with one side to hold the load, and the other affects the load. You can see how pulleys work below.

Two particles A and B, connected by a light inextensible string passed over a smooth fixed peg. The system is held with the string taut and with A and B each at a height of 0.09m above a fixed horizontal plane. It is then released from rest. Particle B becomes stationary when it reaches the plane. Find the tension when both particles are in motion.

Connected particles and pulleys

Let tension be T

Let acceleration be a.

Connected particle concerning pulley projection

Newton's second law will now be applied to each particle.

[equation 1]

[equation 2]

Let's take acceleration due to gravity to be

[equation 1]

[equation 2]

We will add both equations to eliminate T.

We will now substitute a into an equation

## Connected particles on inclined planes

This section is concerned with particles connected with an inextensible string involving inclined planes. Inclined planes are simple machines that are quite different from pulleys. They make work easier by moving objects to a higher elevation and therefore supporting part of the weight of the load on the slope as it moves up. This also means that for the load to move, it will have to be connected to a source of effort moving it over the plane. Let's see how this works.

Two particles A and B of mass 3kg and 4kg respectively are connected by a light inextensible string over a smooth pulley fixed to the edge of the table. Find the tension in the string if particle A is held on a rough plane inclined at 20° to the horizontal, B hangs freely, and the system is released from rest and accelerates at.

Connected particles on inclined planes

We need to remember that the inclined plane has some friction.

Now we need to indicate all the forces acting in our system:

Forces acting on connected particles on an inclined plane

A. We will consider only particle B when we calculate the tension.

Let's take

## Connected Particles - Key takeaways

• Connected particles usually involve pulleys and tension in strings.
• Particles are considered separately in a system that involves multiple particles in motion.
• Newton's second law is often used to help find a force, mass, or acceleration in connected particle problems.
• Light inextensible strings used in connected particles are massless and do not need to be considered in the calculations.

Connected particles can have different masses.

When particles are connected by an inextensible string, they both have the same velocity.

Yes

Friction acts opposite to the direction of applied force.

With the help of Newton's second law, you express most of the activity that goes on in connected particles. You can then find the unknowns.

## Final Connected Particles Quiz

Question

What are connected particles?

They are particles in contact or connected by an inextensible rod or string

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Question

A person of mass 70kg is in a lift of mass 500kg which is attached to a vertical inextensible light cable. The lift is accelerating upwards at 0.6 ms². Find the tension in the string.

T - 70g - 500g = 570 x 0.6

T = 5928 N

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Question

2 boxes A, mass 110kg, and B, mass 190kg, are on the floor of a lift of mass 1700kg. A is on top of B. The lift is supported by a light inextensible cable and descending with a constant acceleration 1.8 ms⁻². Find the tension.

F. = ma

110g + 1700g + 190g - T = (110 + 190 + 1700) x 1.8

T = 16,000 N

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Question

Two particles A and B of masses 10kg and 5kg respectively are connected by a light inextensible string. Particle B hangs directly below particle A. A force of 180 N is applied vertically upwards causing the particles to accelerate. Find the magnitude of the acceleration.

180-15g = 15a

a = 2.2 ms⁻²

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Question

Two particles A and B of masses 10kg and 5kg respectively are connected by a light inextensible string. Particle B hangs directly below particle A. A force of 180 N is applied vertically upwards causing the particles to accelerate. Find the tension in the string.

180-15g = 15a

a = 2.2 ms⁻²

F = ma

T - W = 5 × 2.2

T = 60 N.

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Question

Particles are considered separately in a system that involves multiple particles in motion.

True

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Question

For two bodies in contact with each other, the force each applies to the other is not equal in magnitude and opposite in direction. Is this statement true or false?

False

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